Combinatorial Properties Associated with Generating Functions and Partition Numbers

Combinatorics is the modern mathematics that studies discrete objects, and the problems studied include counting problems, existential problems, enumeration, algorithm problems and optimization problems. In the American Mathematical Society's "Challenges of Mathematics in the 21st Century" large-scale seminar, "Algebraic Methods in Combinatorial Problems" was listed as one of the most challenging directions in pure mathematics. In this century, combinatorics laid the foundation for the computer revolution, because the algorithms of the computer were for discrete objects, and the computer revolution also allowed combinatorics to develop significantly. This paper discusses the fundamentals of combinatorics. A list of future research topics is finally given with clear justifications.